Picture 1

For the transition shown in rice. 1, the given values ​​are: hole diameter d, sides of the base a And b, height N.

Having drawn horizontal projections of the upper and lower bases, i.e. circle and rectangle, connect the vertices of the rectangle with points 0 and 3 of the circle, then construct a frontal projection of the transition.
The lateral surface of such a transition is a combined surface: it consists of four flat triangles marked on Fig.1, but in numbers I And II, and from four conical sections indicated by the number III. The vertices of these four equal conical surfaces lie at the vertices of the rectangle ( points s), and their bases coincide with the circle of the upper base of the transition.

On rice. 1, b the construction of the transition scan began with the construction of triangle I along the side b and height H1, equal to the segment s'ABOUT'(Fig. 1, a). Attached to it on both sides are developments of conical surfaces adjacent to it and tangent to it. III.

Natural lengths of the generatrices S 0 1 0 , S 0 2 0 , S 0 3 0 defined on rice. 1,a by the method of a right triangle and are respectively equal S 0 1 0, S 0 2 0, S 0 3 0. The length of the side l is taken to be equal to the length of the chord of one division of the base. Further construction of the development is clear from the drawing.

The error when replacing an arc with a chord for the corresponding number of divisions will be for the angle α = 30º ~ 1%(with the number of divisions 3), and with the number of divisions equal to four ( α = 22.5º), ~ 0,56% . (Errors associated with the graphical construction of the scan are not taken into account here).

Analytical calculation

The natural lengths of the generators can be calculated using the formula

Formula 1
Where

• Lk - natural length of the corresponding generatrix;
• kα - the angle that determines the position of the projection of the generatrix;
• α = 180º/n when dividing half the base of a circle into n equal parts.

To do this, you need to first determine the value With.

From Figure 1, it is clear that:

Formula 2

Then, the divisions of the circle of the base of the transition must be numbered: put the number 0 at the horizontal projection of the largest generatrix and start counting the angles kα from it.
Size cos kα for the corresponding division can be determined from the table.

Figure 2

For its manufacture, in addition to dimensions H, d and a, you need to set the size e(displacement of the centers of the upper and lower bases). As in the previous case, connecting points s with points 0 And 3 circles, divide the lateral surface of the transition into four conical surfaces, indicated by numbers IV and V, and four triangles labeled I, II, III and tangents to conical surfaces.

The construction of the scan is similar to the previous one and is not shown in the drawing. The only difference is that the developments of the conical elements IV and V will in this case be unequal, and for triangles we will also have three different shapes.

Oblique transition from square to round cross-section

Figure 3

Lateral surface of transition to Fig.3 broken differently than the transitions shown in rice. 1 and 2. The midpoints of the sides of the base a and b (points s and s1) are connected to points 2 of the circle.

As a result of this construction, the lateral surface of the transition will consist of eight triangles I and II tangent to four conical surfaces III And IV. The construction of this development is clear from Fig.3, b. It is similar to the previous ones, but requires more constructions.

Based on materials:
“Technical development of sheet metal products” N.N. Vysotskaya 1968 “Mechanical Engineering”

The main dimensions of a round cone transition (Fig. 129) are: D-diameter of the lower base; d-diameter of the upper base; h - the height of the transition and the opening angle of the transition, which is formed from the intersection of the side faces of the side view of the transition as they continue.

Rice. 129. Development of full and truncated cones

The opening angle in transitions is assumed to be 25-35°, unless there are special instructions in the drawings.

At an opening angle of 25-35°, the transition height is approximately 2 (D-d).

Transitions from round to circular cross-section can have accessible and inaccessible vertices. In the first case, the lateral edges of the lateral type of transition intersect within the sheet when they are continued, in the second case - beyond its boundaries.

The production of a transition from a round to a round section begins with the construction of a development and cutting of individual elements of the transition.

Let's consider techniques for constructing a scan of conical transitions, which are a truncated cone.

A complete cone is the body shown in Fig. 129,a, with base diameter D and top diameter O.

If you roll a cone on a plane around the vertices O, you will get a trace, which will be the development of the cone. The length of the arc constituting the trace of the circle of the base of the cone with diameter D is equal to D, and the radius of size R is equal to the length of the side generatrix of the cone 1.

Unfolding a forward transition with an accessible vertex. If we cut the cone parallel to the base, we will get a truncated cone (Fig. 129, b).

To draw the development of a truncated cone, we construct its side view (ABVG in Fig. 129, c) according to the given for this example diameter of the lower base D = 320 mm, upper base d = 145 mm and height h = 270 mm.

To construct a scan, we continue lines AG and BV until they intersect at point O (Fig. 129, c). If the construction is done correctly, then point O must be located on the center line.

We place a compass at point O and draw two arcs: one through point A and the other through point D; from an arbitrary point B 1 on the lower arc we plot the circumference of the base of the cone, which is determined by multiplying the diameter D by 3.14. Points B 1 and H are connected to vertex O. Figure D 1 B 1 HH 1 will be the development of a truncated cone. To the resulting development we add allowances for folds, as shown in the figure.

The above method of constructing a development of a truncated cone is possible provided that lateral generators AG and BV, as they continue, intersect at an accessible distance from the base of the cone, i.e., at the accessible top of the cone.

Development of a direct transition with an inaccessible vertex. If the diameter of the upper circle of the cone differs little in size from the diameter of the lower circle, then straight lines AG and BV will not intersect within the picture. In such cases, approximate constructions are used to draw the development.

One of the most simple ways An approximate construction of a transition sweep with a small taper is the method of L.A. Laptop.

For example, let us construct a transition scan with a height h = 750 mm, a diameter of the lower base D = 570 mm, and a diameter of the upper base d = 450 mm. To determine the height of the development I, we draw a side view of the transition according to the given dimensions, as shown in Fig. 130, a. The length I of the lateral generatrix of the lateral view of the transition will be the height of the scan. The construction of the sweep of this transition according to the method of L. A. Lapshov (Fig. 130, b) is carried out as follows.

Rice. 130. Development of a circular cross-section transition according to the method of L. A. Lapshov

First, we determine the approximate dimensions of the development, so that when drawing the development, it is possible to correctly position it on the sheets of roofing steel in order to reduce waste and save materials. To do this, we calculate the width of the transition sweep at the lower and upper bases.

The width of the development at the lower base is 3.14 x D = 3.14 x 570 = 1,790 mm, the width of the development at the upper base is 3.14 x d = 3.14 x 450 = 1,413 mm.

Since the scan width longer sheet (1,420 mm), and the height is greater than the width of the sheet (710 mm), then the picture for the transition in length and width will be composed of a sheet with extensions.

The total width of the picture with allowances for folds (single closing fold 10 mm wide and intermediate double fold 13 mm wide) will be equal to 1,790 + 25 + 43 = 1,858 mm.

To construct a scan in the picture we carry out O-O axis"at a distance of approximately 930 mm from the edge (1,858:2). At a distance of 20 mm from the bottom edge of the sheet, we set aside the height of the scan /, the size of which we take from the side view, and find points L and B, as shown in Fig. 130, b Points A and B will be the extreme points of the transition sweep axis. From point B to the left, on a line perpendicular to it, we lay a segment equal to 0.2 (D - d), find point B and connect it with a straight line to point A. In our example, this segment. equal to 0.2 (570 - 450) = 24 mm. This value is a correction for the accuracy of the markings and is determined practically. From points A and B we draw perpendicular lines to the left and plot the values ​​3.14 x d / 8 and 3.14 x D / 8 on them. , i.e. 1/8 of the sweep. We get points 3, 3 1, which we connect with a straight line. In the same way, we build three more times to the left along 1/8 of the transition sweep and get the left half of the transition sweep.

We construct the curves forming the upper and lower sweep arcs using a square and a ruler, as shown in Fig. 130, b.

To the resulting curves we add the width of the flange to the flanges and cut the cutting line with scissors

Then we bend the cut part of the material to the right side of the development according to the template (shaded in the figure) and cut off the excess material. To the resulting development we add an allowance for the longitudinal closing fold.

Development of an oblique transition of circular cross-section. An oblique transition is one in which the centers of the upper and lower bases lie on different axes in one or two planes. The distance between these axes is called the center offset.

Oblique transitions of circular cross-section are used to connect a round fan intake opening with round-section air ducts if their centers lie on different axes.

The development of an oblique transition of a circular cross-section, the surface of which is the lateral surface of a truncated cone, is performed by dividing the entire surface of the oblique transition into auxiliary triangles.

Let us need to construct a development of an oblique transition with a height of H = 400 mm; diameter of the lower base D = 600 mm; diameter of the upper base d = 280 mm; displacement of centers in one plane / = 300 mm.

We build a side view of the oblique transition (Fig. 131, a). To do this, set aside the line AB = 600 mm. From the center of this line - the lower base of the cone - we draw the O 1 -O 1 axis and plot the height H = 400 mm on it. From the top point of height H, draw a horizontal line and mark the offset size on it to the left - 300 mm, find the center O - the upper base. From center O we lay off 140 mm to the left and right - half the diameter of the upper base - and find extreme points C and D. We connect points A and B, B and D with straight lines and get a side view of the oblique transition of the AVGB.

Rice. 131. Development of an oblique transition of a circular cross-section with displacement of the centers of the upper and lower bases in the same plane

To construct a development of half of the transition, we divide its surface into a number of auxiliary triangles.

To do this, we divide the large and small semicircles, each into 6 equal parts, and the division points of the small semicircle are designated by numbers 1", 3", 5", 7", 9", 11" and 13", and the division points of the large semicircle by numbers 1 ", 3", 5", 7", 9", 11" and 13",

Connecting points 1"-1", 1"-3", 3"-3", 3"-5", etc., we get lines 1 1, 2 1, 3 1, 4 1, 5 1, 6 1 , 7 1, 8 1, 9 1, 10 1, 11 1, 12 1 and 13 1, which divide the side surface of half of the transition into auxiliary triangles, on three sides of which there are 1"-1", 1"-3" And 3"-1", etc. - you can construct a development of these triangles.

In these triangles, the only true dimensions on the plan are sides 1"-3", 3"-5", 1"-3", 3"-5", etc.

The sides of the triangles, indicated on the plan by lines under the numbers 1 1, 2 1, 3 1, 4 1, etc., are not true quantities, and therefore are depicted on the plan in abbreviated form (projections).

The true values ​​of these sides will be the hypotenuses of a right triangle, in which one leg is equal to the transition height H, and the other leg is equal to the dimensions of the lines 1 1, 2 1, 3 1, 4 1, 5 1, etc. (Fig. 131, e).

To determine the true values ​​of these lines, we construct a series right triangles with leg a-b equal to H, and legs b - 1 1, b - 2 1, b - 3 1, b - 4 1, etc., equal to lines 1 1, 2 1, 3 1, 4 1, etc. . etc. In these triangles (Fig. 131, c) we find the lengths of the hypotenuses 1, 2, 3, 4, etc.

In order not to obscure the construction, the sizes of lines with odd numbers 1 1, 3 1, 5 1, etc. are set aside on one side leg b-a, and with even numbers 2 1, 4 1, etc. - on the other side of leg b-a.

We construct the development of half of the oblique transition as follows (Fig. 131,d).

We carry out an axial O-O line and on it we lay a line 1"-1", equal to hypotenuse 1. From point 1" with a radius equal to 1"-3", we draw a notch with a compass, and from point 1" with a radius equal to hypotenuse 2, we draw another notch with a compass and find the point 3". Triangle 1" 1" 3" will be the first triangle of the scan. In the same way, a second triangle is attached to it along sides 1"-3" and hypotenuse 3. The remaining triangles are constructed in the same way. The resulting points 1", 3", 5", etc., as well as points 1", 3", 5", etc., are connected by smooth curves, as shown in the figure.

To the resulting contour of the development of half of the oblique transition, allowances for folds and flanges are added.

Using this pattern, the second symmetrical half of the pattern is cut out.

Development of an oblique transition with displacement of the centers of the upper and lower bases in two planes. Suppose we need to construct a scan of an oblique transition having a center offset in the horizontal plane e = 300 mm and a center offset in the vertical plane e 1 = 150 mm; diameter of the lower base D = 700 mm; diameter of the upper base d = 400 mm; height H = 400 mm.

We build a side view, as described above (Fig. 132, a).

Rice. 132. Side view and plan of an oblique transition of a circular cross-section with offset centers of the upper and lower bases in two planes

To build a plan (Fig. 132, b) we proceed as follows.

We build a rectangle with a horizontal side equal to 300 mm (displacement e) and a vertical side equal to 150 mm (displacement e 1). We place the horizontal side of the rectangle between the axes of the upper and lower bases, as shown in Fig. 132, b.

The centers of the upper and lower bases of the oblique transition with an offset in two planes will be located at the vertices of the opposite corners of the rectangle along the diagonal. We draw the O-O axis on this diagonal and build a plan for half of the oblique transition on it. Dividing the plan into separate triangles and constructing a development is performed in the same way as for an oblique transition with an offset in one plane.

After making the transitions, flanges are placed on them, as indicated above.

It is necessary to construct a development of surfaces and transfer the line of intersection of the surfaces to the development. This problem is based on surfaces ( cone and cylinder) with their line of intersection, given in previous problem 8.

To solve such problems in descriptive geometry you need to know:

— the procedure and methods for constructing surface developments;

— mutual correspondence between the surface and its development;

— special cases of constructing developments.

Solution procedurehadachi

1. Note that a development is a figure obtained in
as a result of cutting the surface along any generatrix and gradually unbending it until it is completely aligned with the plane. Hence the development of a right circular cone - a sector with a radius equal to the length of the generatrix and a base equal to the circumference of the base of the cone. All developments are constructed only from natural quantities.

Fig.9.1

— the circumference of the base of the cone, expressed in natural size, is divided into a number of shares: in our case - 10, the accuracy of constructing the scan depends on the number of shares ( Fig.9.1.a);

— we set aside the received shares, replacing them with chords, along the length
arc drawn with a radius equal to the length of the generatrix of the cone l=|Sb|. We connect the beginning and end of the fraction count with the top of the sector - this will be the development of the lateral surface of the cone.

Second way:

— we build a sector with a radius equal to the length of the cone generatrix.
Note that in both the first and second cases the radius is taken to be the extreme right or left generatrice of the cone l=|Sb|, since they are expressed in actual size;

— at the top of the sector we set aside the angle a, determined by the formula:

Fig.9.2

Where r— the radius of the base of the cone;

l— length of the cone generatrix;

360 - a constant value converted into degrees.

We build the base of the radius cone for the development sector r.

2. According to the conditions of the problem, it is required to move the intersection line
surfaces of the cone and cylinder for development. To do this, we use the properties of one-to-one relationship between a surface and its development; in particular, we note that each point on the surface corresponds to a point on the development, and each line on the surface corresponds to a line on the development.

This implies the sequence of transferring points and lines
from the surface to the development.

Fig.9.3

For reaming a cone. Let us agree that the section of the cone surface is made along the generatrix S’ a’ . Then the points 1, 2, 3,…6
will lie on circles (arcs on a development) with radii correspondingly equal to the distances taken along the generatrix S’ A’ from the top S’ to the corresponding cutting plane with points 1’ , 2’, 3’…6’ -| S1|, | S2|, | S3|….| S6| (Fig.9.1.b).

The position of the points on these arcs is determined by the distance taken from the horizontal projection from the generatrix Sa, along the chord to the corresponding point, for example to point c, ac=35 mm ( Fig.9.1.a). If the distance along the chord and arc differ greatly, then to reduce the error you can divide large quantity shares and place them on the corresponding scan arcs. In this way, any points from the surface are transferred to its development. The resulting points will be connected by a smooth curve along the pattern ( Fig.9.3).

For cylinder reaming.

The development of a cylinder is a rectangle with a height equal to the height of the generatrix and a length equal to the circumference of the base of the cylinder. Thus, to construct the development of a right circular cylinder, it is necessary to construct a rectangle with a height equal to the height of the cylinder, in our case 100mm, and a length equal to the circumference of the base of the cylinder, determined by the well-known formulas: C=2 R=220mm, or by dividing the circumference of the base into a number of shares, as indicated above. We attach the base of the cylinder to the upper and lower parts of the resulting development.

Let us agree that the cut is made along the generatrix A.A. 1 (A’ A’ 1 ; A.A.1) . Note that the cut should be made along characteristic (reference) points for more convenient construction. Considering that the development length is the circumference of the base of the cylinder C, from point A’= A’ 1 section of the frontal projection, we take the distance along the chord (if the distance is large, then it is necessary to divide it into parts) to the point B’ (in our example - 17mm) and lay it on a development (along the length of the base of the cylinder) from point A. From the resulting point B we draw a perpendicular (generator of the cylinder). Dot 1 should be on this perpendicular) at a distance from the base taken from the horizontal projection to the point. In our case, the point 1 lies on the symmetry axis of the scan at a distance 100/2=50mm (Fig.9.4).

Fig.9.4

And we do this to find all other points on the scan.

We emphasize that the distance along the scan length to determine the position of the points is taken from the frontal projection, and the distance along the height - from the horizontal, which corresponds to their natural sizes. We connect the resulting points with a smooth curve along the pattern ( Fig.9.4).

In variants of problems when the intersection line breaks up into several branches, which corresponds to a complete intersection of surfaces, the methods for constructing (transferring) the intersection line to a development are similar to those described above.

Section: Descriptive Geometry /

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Cone development. Constructing a cone scan.

Calculation of cone development.

Let's take the vertical and horizontal projections of the cone (Fig. 1, a). The vertical projection of the cone will have the form of a triangle, the base of which is equal to the diameter of the circle, and the sides are equal to the generatrix of the cone. The horizontal projection of the cone will be represented by a circle. If the height of the cone H is given, then the length of the generatrix is ​​determined by the formula:

i.e., like the hypotenuse of a right triangle.

Wrap the cardboard around the surface of the cone. By unfolding the cardboard again into one plane (Fig. 1, b), we obtain a sector whose radius is equal to the length of the generatrix of the cone, and the length of the arc is equal to the circumference of the base of the cone. Full scan the side surface of the cone is performed as follows.

Rice. 1. Cone development:

a - projection; b - scan.

Cone sweep angle.

Taking the generatrix of the cone as the radius (Fig. 1, b), an arc is drawn on the metal, on which a segment of the arc is then laid KM , equal to the circumference of the base of the cone 2 π r. Arc length in 2 π r corresponds to the angle α , the value of which is determined by the formula:

r is the radius of the circle of the base of the cone;

l is the length of the cone generatrix.

The construction of the sweep comes down to the following. Not part of the arc is deposited along the length of the previously drawn arc KM , which is practically impossible, and the chord connecting the ends of this arc and corresponding to the angle α . The magnitude of the chord for a given angle is found in the reference book or indicated on the drawing.

Found points KM connect to the center of the circle. The circular sector obtained as a result of the construction will be the unfolded lateral surface of the cone.